Properties

Degree 1
Conductor 1021
Sign $-0.368 + 0.929i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.572 + 0.819i)2-s + (−0.982 − 0.183i)3-s + (−0.343 + 0.938i)4-s + (−0.412 − 0.911i)5-s + (−0.412 − 0.911i)6-s + (0.989 − 0.147i)7-s + (−0.966 + 0.255i)8-s + (0.932 + 0.361i)9-s + (0.510 − 0.859i)10-s + (−0.659 + 0.751i)11-s + (0.510 − 0.859i)12-s + (−0.602 − 0.798i)13-s + (0.687 + 0.726i)14-s + (0.237 + 0.971i)15-s + (−0.763 − 0.645i)16-s + (−0.993 + 0.110i)17-s + ⋯
L(s,χ)  = 1  + (0.572 + 0.819i)2-s + (−0.982 − 0.183i)3-s + (−0.343 + 0.938i)4-s + (−0.412 − 0.911i)5-s + (−0.412 − 0.911i)6-s + (0.989 − 0.147i)7-s + (−0.966 + 0.255i)8-s + (0.932 + 0.361i)9-s + (0.510 − 0.859i)10-s + (−0.659 + 0.751i)11-s + (0.510 − 0.859i)12-s + (−0.602 − 0.798i)13-s + (0.687 + 0.726i)14-s + (0.237 + 0.971i)15-s + (−0.763 − 0.645i)16-s + (−0.993 + 0.110i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.368 + 0.929i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.368 + 0.929i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $-0.368 + 0.929i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (144, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ -0.368 + 0.929i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5603463833 + 0.8244689475i$
$L(\frac12,\chi)$  $\approx$  $0.5603463833 + 0.8244689475i$
$L(\chi,1)$  $\approx$  0.8299961997 + 0.3713762564i
$L(1,\chi)$  $\approx$  0.8299961997 + 0.3713762564i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.61048347635173744287053828191, −20.814098093988318105310185918378, −19.88178028244954592976653144907, −18.887287040154298056181194902350, −18.261040835969673623926416267968, −17.88380564116632198228534872375, −16.602497958077786984392937210614, −15.682210738754132881625780417075, −15.00687877484254219895241841683, −14.15613763766311397744228163339, −13.47113125834444876727310549610, −12.242742799422333158359263874684, −11.59207304702341943754584136734, −11.230273837117286097565236302435, −10.42826836758184926274012150028, −9.73604259080096603535465368718, −8.439051150740983481944241672491, −7.31524523046665667226679724687, −6.34085649244907358740691832835, −5.58070173438965850028560079699, −4.605680810974098779305181888894, −4.058482155717208499835606142375, −2.77526181371013075670056691710, −1.91926070170243481761837829667, −0.47323592734217211857637474670, 1.0270645367275458085563896947, 2.43356810639670019475679939241, 4.03977487434206909829655698179, 4.86157625325315289223908535084, 5.107288750041927823556911542657, 6.082749033812012741361582631264, 7.45804231084403791711065426588, 7.568568049755735562100147770436, 8.619057419998585706687391581840, 9.74616058103131823517254878242, 10.89016764636809594429870920601, 11.78891894661306488608654230974, 12.36888351576651636983847674870, 13.04684001343124030764329744506, 13.84273690694965896031466624414, 14.995839716950296306104143762027, 15.73024401627084849853261431571, 16.161955482530341010714709574616, 17.24860845546014312551325074583, 17.81302390621732316105254280218, 18.02475680570455606845802960470, 19.70973665147754368485578750475, 20.423294622238969975685158121638, 21.30706180492465381504588840215, 21.98257467822930271114894254251

Graph of the $Z$-function along the critical line